Role of disorder in inducing a BCS-BEC crossover

The role of random onsite and hopping disorder in inducing a Bardeen-Cooper-Schrieffer superconductor to Bose-Einstein condensate (BCS-BEC) crossover is investigated using the Bogoliubov-de Gennes (BdG) method on a negative-U Hubbard model in two dimensions. In the strong disorder limit, the BdG state becomes highly inhomogeneous with the formation of superconducting 'droplets' which are characterized by large pairing amplitudes. The different physical properties calculated for the inhomogeneous state differ significantly from the pure system at all values of electron densities, while we show that a crossover to a BEC state, if at all, is possible at low electron densities. The random onsite disorder supports such a crossover scenario, which is confirmed by observing the chemical potential to slip below the noninteracting band (Leggett criterion). The hopping disorder, however, renders a delocalized phase which allows for overlapping between the pairs, thus preventing the formation of a 'local pair' phase. Also the increase in bandwidth induced by hopping disorder always accommodates the chemical potential inside the band, and hence rules out a crossover phenomenon. However, another variant of the off-diagonal inhomogeneity, i.e. an anisotropic hopping, offers a pro-crossover scenario in the large anisotropy limit.